Inductors and Transformers for Power Electronics pot

However, it is clear that the main cause of the eddy current losses iscaused by the presence of high frequency transverse magnetic field compo-nents.. 1.3.1 Basic Laws for Magnetic Circu

Inductors and Transformers

for Power Electronics

Inductors and Transformers

for Power Electronics

Alex Van den Bossche

Ghent University Gent, Belgium

Vencislav Cekov Valchev

Ghent University Gent, Belgium

Boca Raton London New York Singapore

A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.

The authors try to be accurate and clear, but they cannot guarantee the results or possible interpretations, which might cause direct or indirect injuries, equipment damage, or economic damage by the use of the contents of the book.

Published in 2005 by CRC Press Taylor & Francis Group

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© 2005 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group

No claim to original U.S Government works Printed in the United States of America on acid-free paper

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Library of Congress Cataloging-in-Publication Data

Bossche, Alex van den.

Inductors and tranformers for power electronics / Alex van den Bossche, Vencislav Valchev.

Taylor & Francis Group

is the Academic Division of T&F Informa plc.

To our children

Maxime, Nathan and Laura

Cvetelina and Iasen

This book is mainly intended for designers and users of magnetic components

in power electronics It can also be used for didactical purposes Magneticcomponents such as inductors and transformers constitute together with thecontrol and the semiconductor components, the main parts in the design ofpower electronic converters Some experience teaches that the design of themagnetic parts is still often done by trial and error This can be explained by

a (too) long working-in time for designing inductors and transformers Thedesign has many aspects, such as the magnetic core and winding, eddy cur-rents, insulation, thermal design, parasitic effects, and measurements A lot ofliterature exists concerning those subjects, but the information is spread overmany articles and methods This book is mainly focused on classical methodsand uses numerical tools such as finite element methods in the background

We try to give some overview of the basics and technological aspects of thedesign In the different chapters we also describe analytical approximationsbased on known analytical solutions, but tuned by finite elements In most

of the cases, a sufficient accuracy can be obtained and the results are obtainedalmost instantaneously, even for graphics using many calculation points Afast approximation method is useful as a first step in the design stage, whereasnumerical tools such as finite elements are good in analysis Specific books

on finite elements exist and the description will not be repeated here.Some basic introduction on magnetic principles and materials are given

in Chapter 1

Today power electronics use quite a high switching frequency.Simple rules

of thumb such as that “the eddy copper current losses are always negligiblewhen the diameter of the wire is smaller than the penetration depth” are nottrue However, it is clear that the main cause of the eddy current losses iscaused by the presence of high frequency transverse magnetic field compo-nents This is the base of the fast design method in Chapter 2 The method

is further improved using some corrections for other effects and is embedded

in a decision flow chart of a design procedure More insight and betteraccuracy is provided in the other chapters We invite the readers to let themguide by the contents of the book to their specific subjects of interest.The chapters in the book are organized in a quite independent way withrespective local appendices and references The general appendices at theend provide information that is not linked to a specific chapter and can beused independently

This work can be seen as complementary information to books on powerelectronic circuits Different levels of complexity are proposed depending onthe available time, the desired accuracy, and the mathematical level of thedesigner

We want to thank several institutions, that permitted the research and theachievement of this book: DWTC and FWO – Belgium; NATO ResearchProgram; BOF in Ghent University; Fellowships of scientific exchangebetween Belgium and Bulgaria; E.E.C Tempus and Socrates exchangeprograms

The authors are also grateful to the department heads Prof Dr ir JanMelkebeek of the Ghent University Electrical Energy Laboratory and Prof

Dr Eng Dimitar Yudov, who supported us

We want also to acknowledge the companies such as Philips, Tyco, Inverto,Barco, Fabricom in Belgium and Struna in Bulgaria The opportunity todesign for them induced industrial realism

Many collaborators did a wonderful job while reading and giving tions of improvements and encouragements to the fulfillment of this book

sugges-About the Authors

Alex P M Van den Bossche received the M.S and the Ph.D degrees fromthe University of Ghent, Belgium in 1980 and 1990 respectively He hasworked there at the Electrical Energy Laboratory Department, EESA He hasbeen engaged in research and published articles in the field of electricaldrives and power electronics concerning various converter types, drives andvarious aspects of magnetic components and materials His interests are also

in renewable energy conversion Since 1993, he has been a full professor atthe same university He is a senior member of the IEEE (M’99S’03)

Vencislav V Valchev received the M.Sc and Ph.D degrees in electricalengineering from the Technical University of Varna, Bulgaria in 1987 and

2000, respectively Since 1988 he has been with the Department of Electronics,Technical University of Varna, where he has been a lecturer His researchinterests include power electronics, soft switching converters, resonant con-verters, and magnetic components for power electronics, renewable energyconversion

Dr Valchev had a cumulated common research period of about four years

in the Electric Energy Laboratory research group in Ghent University,Belgium

The symbols do mainly follow the standard ISO 31-11

Concerning upper and lower cases we try to keep the following conventions:Voltage and current:

Time dependent values of voltage and current are denoted by lowcases (v, i)

RMS values are capitals without index for sinusoidal waveforms.The indexrms is mentioned explicitly for RMS values of non-sinusoidalwaveforms

Field quantities such as H and B are always written in capital case, the contextshows what it is e.g B p = is the peak value of the induction B(t) is the valuedepending on time

Matrices and vectors are written in bold

Variables are written in italic

Functions, operators, universal constants are non-italic

Complex variables are underlined if confusion is possible

Blanks are used as multiplication

We did split the nomenclature in variables, subscripts, superscripts, constantsand frequently used abbreviations The specific combination of variables withsubscripts is defined in the respective chapters at their first appearance

Variables

xiv Nomenclature

Nomenclature xv

for coefficients

max maximum

tip tip (top or bottom of foil)

Frequently Used Abbreviations

−1

Table of Contents

1 Fundamentals of Magnetic Theory

1.1 Basic Laws of Magnetic Theory

1.1.1 Ampere’s Law and Magnetomotive Force

1.1.2 Faraday’s Law and EMF

1.1.3 Lenz’s Law and Gauss’s Law for Magnetic Circuits1.2 Magnetic Materials

1.3.1 Basic Laws for Magnetic Circuits

1.3.2 Inductance

1.3.2.1 Flux Linkage1.3.2.2 Inductance: Definitions1.3.2.3 Inductance: Additional Considerations1.3.2.4 Self-inductance and Mutual Inductance1.3.3 Transformer Models

1.3.3.1 Ideal Transformer1.3.3.2 Practical Transformer1.3.4 Magnetic and Electrical Analogy

References

2 Fast Design Approach Including Eddy Current Losses

2.1 Fast Design Approach

2.1.1 Non-Saturated Thermal Limited Design

Step 1) Choose a Core Material and SizeStep 2) Calculate the Heat Dissipation Capability P h

Step 3) Copper Loss/Core Loss RatioStep 4) Calculate the Specific Core Losses P fe ,sp

Step 5) Find the Peak Induction B p,g from

Graphical DataStep 6) Check if the Peak Induction B p is Higher Than

the Saturation Value B sat

Symmetrical WaveformsAsymmetrical WaveformsStep 7) Calculate the Winding Turns N i

Step 8) Distribute Allowed Total Copper Losses P h,cu

Among the WindingsStep 9) Determine Wire Diameter d i Step 10) Calculate the Actual Copper Losses P cu

I) Ohmic Copper LossesII) Low-Frequency Transverse Field Eddy Current Losses

III) Wide Frequency Eddy Current Losses

IV) Total Copper Losses

Step 11) Check if the Copper Losses P cu are Lower Than

the Allowed Copper Dissipation P h,cu

Step 12) Is Improvement Possible?

Step 12a) Optimize the Diameter and Winding

Arrangement

I) TransformersII) InductorsStep 13) Check the Copper Filling FactorStep 13a) Choose a Larger Core

Step 14) Check if the Chosen Core Size in Step 1)

is not Too HighStep 14a) Choose a Smaller CoreStep 15) Calculate the Total Air Gap Length Σl g

2.1.2 Saturated Thermally Limited Design

Step 1’) Find the Peak-to-Peak Induction B pp

Step 2’) Choose a Core, Material, and Size

Step 3’) Find the Core Losses P fe from Graphical Data Step 4’) Find the Heat Dissipation Capability

P h of the Component

Step 5’) Check the Ratio P fe /P h

Step 6’) Estimate the Allowed Copper

Dissipation Capability2.1.3 Signal Quality Limited Design

2.2 Examples

2.2.1 Non-Saturated Thermally Limited

Design Example2.2.1.1 Design Steps

Conclusions:

2.2.1.2 Improvements of the Design 2.2.1.3 Measuring and Validation of the Design2.2.2 Saturated Thermal Limited Design Example

2.2.2.1 Design Procedure

Equation Approach2.2.2.2 Measurements and Validation

of the Eddy Current Losses 2.3 Conclusions

Appendix 2.A.1 Core Size Scale Law for Ferrites

in Non-Saturated Thermal Limited Design

Appendix 2.A.2 Eddy Current Losses for Wide Frequency

2.A.2.1 Approximation of k c

2.A.2.2 Transformers

2.A.2.2.1 Direct Calculations2.A.2.2.2 Graphical Transformer Method2.A.2.3 Inductors

2.A.2.3.1 Direct Calculations2.A.2.3.2 Graphical Inductor MethodAppendix 2.A.3 Mathcad Example Files

References

3 Soft Magnetic Materials

3.1 Magnetic Core Materials

3.1.1 Iron-Based Soft Magnetic Materials

3.1.1.1 Laminated Cores3.1.1.2 Powdered Iron and Carbonyl Iron Cores

Powdered Iron Carbonyl Iron 3.1.1.3 Amorphous Alloys

Production Process and Microstructure CharacteristicsMagnetic Properties

ApplicationsShapes3.1.1.4 Nanocrystalline Magnetic Materials

Production Process and Microstructure CharacteristicsMagnetic Properties

Temperature BehaviorShapes

Applications3.1.2 Ferrites

Production Process and Microstructure CharacteristicsMagnetic Properties

Low Induction Level (Signal Level) Parameters

High Induction Level (Power Level) Parameters

Shapes3.2 Comparison and Applications of the Core Materials

in Power Electronics

3.3 Losses in Soft Magnetic Materials

3.3.1 Simplified Approach for Laminated Steel Cores

3.3.2 Hysteresis Losses

3.3.3 Eddy-Current Losses

3.3.3.1 Eddy Current Losses in Laminated Cores

Low Frequency Approximation

of Eddy Current Losses in Laminated Cores

3.3.3.2 Eddy Current Losses in Laminated

Cores at Arbitrary Frequencies3.3.4 Anomalous (Residual, Excess) Losses

3.4 Ferrite Core Losses with Non-Sinusoidal

Voltage Waveforms

3.4.1 Identification of the Steinmetz Equation

3.4.2 Natural Steinmetz Extension for Ferrite Core Losses

with Non-Sinusoidal Voltage Waveforms3.5 Wide Frequency Model of Magnetic Sheets Including

Hysteresis Effects

3.5.1 Constant Loss Angle Impedance

3.5.2 Transmission Line Approach with Constant

Loss Angle Material3.5.3 Wide Frequency Complex Permeability Function

3.5.4 Real, Reactive, and Apparent Power

3.5.5 Dependence on Saturation Level

3.5.6 Wide Frequency Model Curves of Typical Materials

3.5.6.1 Silicon Steel3.5.6.2 Nanocrystalline Material 3.5.6.3 Wide Frequency Model for FerritesAppendix 3.A Power and Impedance of Magnetic Sheets

4.1.3 Wires with Rectangular Cross Section

4.1.4 Litz Wires

4.2 Wire Length

4.2.1 Circular Coil Formers

4.2.2 Rectangular Coil Formers

4.3 Physical Aspects of Breakdown

4.3.1 Breakdown Voltage in Air

4.3.2 Breakdown Voltage in Solid Insulation Material

4.3.3 Corona Discharge

4.4 Insulation Requirements and Standards

4.4.1 Basic, Supplementary, and Reinforced

Insulation4.4.2 Standard Insulation Distances

4.4.2.1 Clearance4.4.2.2 Creepage Distance 4.4.3 Electric Strength Tests

4.4.4 Leakage Currents

4.5 Thermal Requirements and Standards

4.5.1 Thermal Evaluation of Insulation Materials and Systems

4.5.2 Requirements and Standards for Inductive

(Magnetic) Modules4.5.3 Standards for Wires

Bare Material DiameterEnamel Thickness Resistance Per Meter Thermal Classes of Magnet Wires 4.6 Magnetic Component Manufacturing Sheet

Coupling Air GapsImpregnatingPartially Filled Layer References

5 Eddy Currents in Conductors

5.1 Introduction

Current Power Electronics Needs

Skin Effect

Proximity Effect

Air Gap Effects

Eddy Current Losses in Conductors

5.2 Basic Approximations

5.2.1 Low Frequency Approximation

5.2.2 High Frequency Approximation

5.2.3 Superposition of Losses

5.2.4 Wide Frequency Approximation

5.3 Losses in Rectangular Conductors

5.3.1 Exact Solution For a Current Carrying Rectangular

Conductor in a Transverse Field 5.3.2 Low Frequency Approximation

5.3.2.1 Current Carrying Conductor Without

Transverse Field 5.3.2.2 Conductor Without Current in a

Transverse Field 5.3.3 High Frequency Approximation

5.3.3.1 Ideal Case

5.3.4 Spaced Conductors

5.3.4.1 Classical Approach 5.3.4.2 Low Filling Factor and High Frequency 5.4 Quadrature of the Circle Method for Round Conductors

5.4.1 Equivalent Rectangle Principle

5.4.2 Adapted Equations

5.4.3 Low Frequency Approximation

Accuracy of Dowell Method5.4.4 Improved Quadrature of the Circle Method

5.4.5 Discussion of Quadrature of the Circle Methods

Conclusions for Classical Dowell Method Conclusions for IQOC Method

5.5 Losses of a Current Carrying Round Conductor

in 2-D Approach

5.5.1 Exact Solution

5.5.2 Low and High Frequency Approximation

5.5.3 Wide Frequency Approximation

5.6 Losses of a Round Conductor in a Uniform

Transverse AC Field

5.6.1 Exact Solution

5.6.2 Low Frequency Approximation

5.6.3 High Frequency Approximation

5.6.4 Wide Frequency Approximation

5.6.5 Discussion

5.7 Low Frequency 2-D Approximation Method

for Round Conductors

5.7.1 Direct Integration Method for Round Wires

5.7.2 Three-Field Approximation

5.7.3 Solution in a Magnetic Window Using Mirroring

5.7.4 Suppression of the First Infinite Sum

5.8 Wide Frequency Method for Calculating Eddy Current

Losses in Windings

5.8.1 High Frequency Effect of Other Wires,

Using Dipoles5.8.2 Wide Frequency Method, Tuning with Finite

Element Solutions5.8.2.1 A Wire in a Transverse Field5.8.2.2 A Wire in a Half Layer

Conclusions of the Comparisons5.8.2.3 Losses in the General Case of a

Transformer Winding5.8.2.4 Losses in an Inductor Winding 5.8.3 High Frequency, High Filling Factor Relations

5.8.4 Summary of the Wide Frequency Method

5.8.5 Comparison of Analytically Based Methods

5.8.5.1 Low Frequency Methods5.8.5.2 Wide Frequency Method and Quadrature

of Circle Methods 5.9 Losses in Foil Windings

5.9.1 Homogenous Field Parallel to the Foil

5.9.2 Induced Losses by Air Gaps

5.9.2.1 Analytical Modeling5.9.3 Tip Currents in Foil Conductors

Foil Inductors Foil TransformersConclusions Concerning Tip Currents5.9.4 Conclusions for Foil Windings

5.10 Losses in Planar Windings

Advantages of the Planar CoresLosses in Planar Magnetic Components Specifics

Appendix 5.A.1 Eddy Current 1-D Model for

Rectangular Conductors

5.A.1.1 Basic Derivations

5.A.1.2 Single Conductor in a Slot

5.A.1.3 Superimposed Rectangular Conductors in a Slot

5.A.1.4 Taylor Expansion and Low Frequency

Approximation for Superimposed RectangularConductors in a Slot

5.A.1.5 Approximation for Rectangular Conductors

with Air5.A.1.5.1 Classical ApproachAppendix 5.A.2 Low Frequency 2-D Models for Eddy

Current Losses in Round Wires

5.A.2.1 Low Frequency Approach

5.A.2.2 Defining a 2-D Winding Arrangement

5.A.2.3 Eddy Current Losses by The Direct

Integration Method5.A.2.4 The Proposed Three Orthogonal Fields Method

5.A.2.4.1 The Field of the Conductor5.A.2.4.2 The Transverse Field5.A.2.4.3 The Hyperbolic Field5.A.2.4.4 Residual Field5.A.2.4.5 Eddy Current Losses by the Three

Orthogonal Fields5.A.2.5 Validation of the Proposed 3-Field Approximation

5.A.2.6 Extension of the Obtained Solution

Appendix 5.A.3 Field Factor For Inductors

5.A.3.1 2-D Analytical Approximation of the

Field Factor k F

5.A.3.2 Simplified Approach

5.A.3.3 Parallel and Perpendicular Components of k F

References

6 Thermal Aspects

6.1 Fast Thermal Design Approach (Level 0 Thermal Design)

6.1.1 Specific Dissipation p for Ferrites

6.1.2 Conclusion About Level 0 Thermal Design

6.2 Single Thermal Resistance Design Approach

(Level 1 Thermal Design)

6.3 Classic Heat Transfer Mechanisms

6.3.1 Conduction Heat Transfer

6.3.2 Convection Heat Transfer

6.3.2.1 Natural and Forced Convection

6.3.2.2 Convection Heat Transfer Coefficient h c

6.3.3 Radiation Heat Transfer

6.4 Thermal Design Utilizing a Resistance Network

Level 2 Thermal Design6.4.1 Thermal Resistances

6.4.2 Finding Temperature Rise

6.5 Contribution to Heat Transfer Theory of Magnetic Components

6.5.1 Practical Experience

6.5.2 Precise Expression of the Natural Convection

Coefficient h c

6.5.2.1 Derivation of Convection Coefficient h c

6.5.2.2 Dependencies of h c on the Parameter L and

on the Position and Shape6.5.3 Forced Convection

6.5.3.1 Classical Approach6.5.3.2 Adapted Approach6.5.4 Relationship with Thermal Resistance Networks

6.6 Transient Heat Transfer

6.6.1 Thermal Capacitances in Magnetic Components

6.6.2 Transient Heating

6.6.3 Adiabatic Loading Conditions

6.7 Summary

Appendix 6.A Accurate Natural Convection Modeling

for Magnetic Components

6.A.1 Experimental Set Up

6.A.2 Thermal Measurements with the Box-Type Model

6.A.3 Thermal Measurements with the EE Transformer

Type Model6.A.3.1 Thermal Measurements at an Ambient

Temperature of 25°C 6.A.3.2 Thermal Measurements at an Ambient

Temperature of 60°C

6.A.4 Derivation of an Accurate Presentation of the

Convection Coefficient h c

6.A.5 Comparison of the Experimental Results and Proposed Thermal Modeling

References

7 Parasitic Capacitances in Magnetic Components

7.1 Capacitance Between Windings: Inter Capacitance

7.1.1 Effects of the Inter Capacitance

7.1.2 Calculating Inter Capacitances and the

Equivalent Voltage 7.1.3 Measuring Inter Capacitances

7.2 Self-Capacitance of a Winding: Intra Capacitance

7.2.1 Effects of Intra Capacitance

7.2.2 Calculating Intra Capacitances of a Winding

7.2.3 Measuring Intra Capacitances of Windings

7.2.3.1 Single Parasitic Capacitance Model7.2.3.2 Model with a Parasitic Capacitance for

Each Winding7.3 Capacitance Between the Windings and the

Magnetic Material

7.4 Practical Approaches for Decreasing the Effects

of Parasitic Capacitances

7.4.1 Low Intra-Capacitance Windings

7.4.2 Decreasing the Effects of the Inter Capacitance

8.3 Typical Ferrite Inductor Shapes

8.4 Fringing in Wire-Wound Inductors with Magnetic Cores

8.4.1 Center Gapped, Spacer and Side Gapped Inductors

8.4.2 Simplified Approach to the Center Gapped Inductors

8.4.3 Improved Approximation for Fringing Permeances

of Gapped Inductors8.4.3.1 Fringing Coefficients8.4.3.2 Equivalent Surface8.4.3.3 Single and Multiple Air Gap Cases

8.5 Eddy Currents in Inductor Windings

8.5.1 Referring to Described Methods

8.5.2 Multiple Air Gap Inductors

8.5.3 Avoiding Winding Close to the Air Gap

8.6 Foil Wound Inductors

8.6.1 Foil Inductor—Ideal Case

8.6.2 Single and Multiple Air Gap Design

in Foil Inductors 8.6.3 Eddy Current Losses in Foil Windings

of Gapped Inductors8.6.4 Planar Inductors

8.7 Inductor Types Depending on Application

8.7.1 DC Inductors

8.7.2 HF Inductors

8.7.3 Combined DC-HF Inductor

8.7.3.1 Classical Solutions8.7.3.2 Special, Combined Design: Litz

Wire–Full Wire Inductor Winding8.7.3.3 Analytical Modeling of the Combined

Full-Wire–Litz-Wire Inductor8.8 Design Examples of Different Types of Inductors

8.8.1 Boost Converter Inductor Design

8.8.2 Coupled Inductor Design

8.8.3 Flyback Transformer Design

8.A.1 Fringing Coefficients For Gapped-Wire-Wound Inductors

8.A.1.1 Basic Cases

8.A.1.1.1 Basic Case 18.A.1.1.2 Basic Case 28.A.1.1.3 Basic Case 38.A.1.1.4 Basic Case 48.A.1.2 Symmetrical Cases

8.A.1.2.1 Case 1s8.A.1.2.2 Case 2s8.A.1.2.3 Case 3s8.A.1.2.4 Case 4s8.A.1.3 Application to Gapped Rectangular Cores

8.A.1.4 Application to Center Gapped Rectangular Cores

8.A.1.5 Application to Center Gapped Round Cores

8.A.2 Analytical Modeling of Combined

Litz-Wire–Full-Wire Inductors

8.A.2.1 Example of a Combined

Litz-Wire–Full-Wire Inductor8.A.2.2 Experimental Results

8.A.2.3 Conclusion

References

9.3.1 Leakage Inductance of Concentric Windings

9.3.2 Leakage Inductance of Windings

in Separate Rooms 9.3.2.1 General Case9.3.2.2 Axis-Symmetrical Case9.3.3 Leakage Inductance in T, L and M Models

of Transformers

9.3.3.1 T Transformer Model 9.3.3.2 L Transformer Model 9.3.3.3 M Transformer Model

9.4 Using Parallel Wires and Litz Wires

9.4.1 Parallel Wires

9.4.1.1 Low Frequency Case: d < 1.6 δ

9.4.1.2 High Frequency Case: d > 2.7δ9.4.2 Parallel Windings Using Symmetry in the

Magnetic Path 9.4.3 Using Litz Wire

9.4.3.1 Example in the Low-Frequency

Approximation9.4.4 Half Turns

10.2 Loss Minimization in the General Case

10.3 Loss Minimization Without Eddy Current Losses

10.3.1 Constant Copper Volume

10.3.2 Constant Wire Cross Section

10.3.3 Equal Core and Copper Surface Temperatures

10.4 Loss Minimization Including Low-Frequency

Eddy Current Losses

10.4.1 Constant Copper Wire Cross Section

10.4.2 Constant Copper Wire Volume

10.4.3 Variable Wire Cross Section and Number of Turns

10.4.4 More General Problems with Eddy Currents

11.2.2 PT100 Thermistor Temperature Measurement

11.2.3 NTC Thermistor Temperature Measurement

11.2.4 Glass Fiber Optic Temperature Measurement

11.2.5 Infrared Surface Temperature Measurement

11.2.6 Thermal Paint and Strips

11.2.7 Winding Resistance Measurement Method

11.3 Power Losses Measurements

11.3.1 Circuit Wattmeter Measurement

11.3.2 Oscilloscope Measurements

11.3.1.1 Example of the Accuracy Problem

in Oscilloscope Measurement11.3.2 Impedance Analyzers and RLC Meters

11.3.2.1 Impedance Analyzers11.3.2.2 RLC Meters

11.3.3 Q-factor Test of LC Networks

11.3.4 Power Loss Estimation by Thermal Resistance

11.3.5 Calorimetric Power Loss Measurement

11.3.5.1 Inertia Calorimeter11.3.5.2 Flow Calorimeter

11.3.5.2.1 Principle of Operation11.3.5.2.2 Accuracy of Flow Calorimeters11.3.5.2.3 Practical Flow Calorimeter11.3.5.2.4 Conclusions

11.4 Measurement of Inductances

11.4.1 Measurement of the Inductance of an Inductor

11.4.2 No Load Test of Transformers

11.4.3 Short Circuit Test

11.4.4 Measurement of the Inductances in Transformers

11.4.5 Measurement of Low Inductances

11.5 Core Loss Measurements

11.5.1 Classical Four-Wire Method

11.5.2 Two-Wire Method

11.5.2.1 Osciloscope Based Measurement11.5.2.2 Wide Band Current Probe

11.5.2.3 Corresponding Voltage Probe11.5.2.4 Flux Measurement Probe11.5.3 Practical Ferrite Power Loss Measurement

Set Up 11.6 Measurement of Parasitic Capacitances

11.6.1 Measurement of Capacitance Between Windings

11.6.2 Measurement of the Equivalent Parallel

Capacitance of a Winding 11.7 Combined Measuring Instruments

A.2.1 Discontinuous Waveforms

A.2.2 Repeating Line Waveforms

A.2.3 Waveforms Consisting of Different Repeating

Line PartsA.3 RMS Values of Common Waveforms

A.3.1 Sawtooth Wave, Fig A.4

A.3.2 Clipped Sawtooth, Fig A.5

A.3.3 Triangular Waveform, No DC Component, Fig A.6

A.3.4 Triangular Waveform with DC Component, Fig A.7

A.3.5 Clipped Triangular Waveform, Fig A.8

A.3.6 Square Wave, Fig A.9

A.3.7 Rectangular Pulse Wave, Fig A.10

A.3.8 Sine Wave, Fig A.11

A.3.9 Clipped Sinusoid, Full Wave, Fig A.12

A.3.10 Clipped Sinusoid, Half Wave, Fig A.13

A.3.11 Trapezoidal Pulse Wave, Fig A.14

Appendix B Magnetic Core Data

B.1 ETD Core Data (Economic Transformer Design Core)

B.2 EE Core Data

B.3 Planar EE Core Data

B.4 ER Core Data

B.5 UU Core Data

B.6 Ring Core Data (Toroid Core)

B.7 P Core Data (Pot Core)

B.8 PQ Core Data

B.9 RM Core Data

B.10 Other Information

Appendix C Copper Wires Data

C.1 Round Wire Data

C.2 American Wire Gauge Data

C.3 Litz Wire Data

Appendix D Mathematical Functions

References

Fundamentals of Magnetic Theory

This chapter gives a brief review of the basic laws, quantities, and units ofmagnetic theory Magnetic circuits are included together with some examples.The analogy between electric and magnetic circuits and quantities is pre-sented Hysteresis and basic properties of ferromagnetic materials are alsodiscussed The models of the ideal transformers and inductors are shown

1.1 Basic Laws of Magnetic Theory

The experimental laws of electromagnetic theory are summed up by the well equations In 1865, after becoming acquainted with the experimental results

Max-of his fellow Englishman Faraday, Maxwell gave the electromagnetic theory acomplete mathematical form We will present specific parts of the Maxwellequations: Ampere’s law, Faraday’s law, and Gauss’s law, which together withLenz’s law are the basis of magnetic circuit analysis These are the laws that areuseful in the design of magnetic components for power electronics

When an electrical conductor carries current, a magnetic field is inducedaround the conductor, as shown in Fig 1.1 The induced magnetic field is

characterized by its magnetic field intensity H The direction of the magnetic field intensity can be found by the so-called thumb rule, according to which,

if the conductor is held with the right hand and the thumb indicates thecurrent, the fingers indicate the direction of the magnetic field

The magnetic field intensity H is defined by Ampere’s law According to

Ampere’s law the integral of H [A/m] around a closed path is equal to the

total current passing through the interior of the path (note that a line above

a quantity denotes that it is a vector):

(1.1)

H⋅ l=∫J⋅ S

S l

H is the field intensity vector [A/m]

dl is a vector length element pointing in the direction of the path l [m]

J is the electrical current density vector [A/m2]

dS is a vector area having direction normal to the surface [m2]

l is the length of the circumference of the contour [m]

S is the surface of the contour [m2]

If the currents are carried by wires in a coil with N turns, then

(1.2)

where

i is the current in the coil

N is the number of the turns

The terms and Ni in Equation (1.2) are equivalent to a source called

magnetomotive force (MMF), which is usually denoted by the symbol F [A ⋅ turns]

Note that the number of turns N does not have dimension, but the value Ni

is an actual MMF and not a current According to Equation (1.1) the net MMF

around a closed loop with length l c is equal to the total current enclosed bythe loop Applying Ampere’s law to Fig 1.1 we obtain

Illustration of Ampere’s law The MMF

around a closed loop is equal to the sum of

the positive and negative currents passing

through the interior of the loop.

Surface S with area Ac

Total current i Total density J

H l

m is a specific characteristic of the magnetic material termed permeability

m0 is the permeability of free space, a constant equal to 4π× 10–7 H/m

m r is the relative permeability of the magnetic material

The value of m r for air and electrical conductors (e.g., copper, aluminum)

is 1 For ferromagnetic materials such as iron, nickel, and cobalt the value

of m r is much higher and varies from several hundred to tens of thousands

The magnetic flux density B is also called magnetic induction and, for simplicity, in this book we will use the term induction for this magnetic

quantity The vector B is the surface density of the magnetic flux The scalar

value of the total magnetic flux Φ passing through a surface S is given by

(1.5)

If the induction B is uniform and perpendicular to the whole surface area

A c, then the expression in Equation (1.5) results in

We have to mention that the expression given by Equation (1.1) is notcomplete; there is a term missing in the right-hand side The missing term,

which is a current in fact, is called displacement current and was added to the

expression by Maxwell in 1865 The full form of the law is

(1.7)

where

e is the permittivity of the medium

E is the electric field

Maxwell’s correction to Ampere’s law is important mainly for frequency applications with low current density In magnetic componentsfor power electronics the expected current density is of the order of at least

high-J = 106 A/m2 In all normal applications the second term on the right-handside of Equation (1.7) (the Maxwell’s correction) is almost surely not morethan 10 A/m2, and can be neglected Exceptions are the currents in capacitors,currents caused by so-called parasitic capacitances, and currents in trans-mission lines This conclusion allows us to use the simplified expression inEquation (1.1) in power electronics magnetic circuit analysis, an approachcalled the quasi-static approach

1.1.2 Faraday’s Law and EMF

A time-changing flux Φ(t) passing through a closed loop (a winding) erates voltage in the loop The relationship between the generated voltage v(t)

gen-and the magnetic flux Φ(t) is given by Faraday’s law According to Faraday’s law the generated voltage v(t) is

con-convention The positive senses of B, dl, dS, and the generated electromotive

force (EMF) are shown by arrows in Fig 1.2

Faraday’s law is valid in two cases:

• A fixed circuit linked by a time-changing magnetic flux, such as atransformer

• A moving circuit related to a time-stationary magnetic flux in a waythat produces a time-changing flux passing through the interior ofthe circuit

Rotating electrical machines generate EMF by the latter mechanism

Lenz’s law states that the voltage v(t) generated by a fast time-changing

magnetic flux Φ(t) has the direction to drive a current in the closed loop,

which induces a flux that tends to oppose the changes in the applied flux

Φ(t) Figure 1.3 shows an example of Lenz’s law

FIGURE 1.2

Illustration of Faraday’s law The voltage v(t)

induced in a closed loop by a time-changing flux

Φ(t) passing the loop (generator convention).

B(t)

dS dl

EMF + −

Lenz’s law is useful for understanding the eddy current effects in magneticcores as well in the coil conductors The eddy currents are one of the majorphenomena causing losses in magnetic cores and in coil conductors.

Gauss’s law for magnetic circuits states that for any closed surface S with

arbitrary form the total flux entering the volume defined by S is exactly equal

to the total flux coming out of the volume This means that the total resultingflux through the surface is zero:

The relative permeability m r of diamagnetic and paramagnetic materials is

close to unity The values of B and H are linearly related for both materials.

Diamagnetic materials have a value of m r less than unity, which means that theytend to slightly exclude magnetic field, that is, a magnetic field intensity isgenerally smaller in a diamagnetic material than it would be in a paramag-netic material under the same conditions The atoms of diamagnetic materials

FIGURE 1.3

Illustration of Lenz’s law in a closed winding The applied flux Φ(t) induces current i(t), which

generates induced flux Φ ι(t) that opposes the changes in Φ(t).

Applied flux Φ (t) Induced flux Φi(t)

Induced current i (t) Closed loop

B⋅ S=

S

do not have permanent magnetic moments Superconductors are a specific

class of diamagnetic materials In these materials there are macrocurrentscirculating in the structure These currents oppose the applied field and as

a result the material excludes all exterior fields Paramagnetic materials have a value of m r greater than unity, and they are slightly magnetized by an applied

magnetic field Ferromagnetic materials are characterized by values of m r muchhigher than unity (10–100,000) [1] For the design of magnetic componentsfor power electronics, the third type of materials, the ferromagnetic mate-rials, are of real importance, especially ferromagnetic ceramics and metals

Comparison of B-H relation of different types of magnetic materials is shown

in Fig 1.4

To understand ferromagnetic materials we will start with the magneticmoments of atoms and the structure of metals Each electron possesses anelectrical charge and its own magnetic (spin) moment Besides the spin, eachelectron of the atom has another magnetic moment, a so-called orbitalmoment, caused by its rotation around the nucleus In the atoms of manyelements the electrons are arranged in such a way that the net atomic moment

is almost zero Nevertheless, the atoms of more than one-third of the knownelements possess a magnetic moment Thus, every single atom of theseelements has a definite magnetic moment as a result of the contributions ofall of its electrons This magnetic moment can be associated with an atomicmagnet

In metals there is an interaction between the atoms, which defines themagnetic properties of the total structure In most cases the atomic moments

in the crystal are inter-coupled by coupling forces If the atomic momentsare arranged in parallel with crystal lattice sites, then the moments of the

individual atoms are summed up resulting in the ferromagnetic effect The

coupling forces in the ferromagnetic materials of technical interest are strongand at room temperature almost all atomic magnets are parallel-aligned Thealignment of the atomic magnets does not occur in the entire structure, butonly within certain regions These regions of alignment of the atomic mag-

nets are called ferromagnetic domains or Weiss domains In polycrystalline

FIGURE 1.4

Magnetization curves for different types

of magnetic materials The scale of the

magnetization curve of ferromagnetic

ma-terials is much higher.

B

Ferromagnetic

Paramagnetic Free air Diamagnetic

H

materials they usually have a laminar pattern The size of the domains variesconsiderably, from 0.001 mm3 to 1 mm3 Each domain contains many atomsand is characterized by an overall magnetic moment, as a result of thesumming of the atomic magnets The directions of the domain magneticmoments in an unmagnetized crystal are not completely random among allavailable directions The domain magnetic moments are oriented so as tominimize the total external field, and in that way to keep the energy content

as low as possible To follow this rule, adjacent domains have oppositemagnetic moments, as shown in Fig 1.5 The net external field is reduced

additionally by so-called closure domains, shown in Fig 1.5.

In every crystal the domains are divided from each other by boundaries,

so-called domain walls or Bloch walls Across the domain walls the atomic

magnetic moments reverse their direction, as shown in Fig 1.6

The described mechanism of summing the atomic magnetic moments,resulting in spontaneous magnetization of the domains in ferromagnetic

materials, is valid until a specific temperature, called the Curie temperature

T C The value of T C is clearly defined for every material If the temperature

of the material is increased above that value the thermal oscillations of theatomic magnets increase significantly and overcome the coupling forces thatmaintain the alignment of the atomic magnets in the domains The finaleffect disturbs the alignment of magnetic moments of adjacent atoms When

a ferromagnetic material is heated above its Curie temperature T C, its netic properties are completely changed and it behaves like a paramagnetic

mag-material The permeability of the material drops suddenly to m r ≈ 1, and bothcoercivity and remanence become zero (the terms coercivity and remanencewill be discussed in the next section) When the material is cooled, thealignment of the atomic magnets in the domains will recover, but the mag-netic moments of the domains will be orientated randomly to each other

FIGURE 1.5

Orientation of domain magnetic moments in

the structure of unmagnetized iron.

Thus, the total external field in the structure will be zero This means that

heating a ferromagnetic material above T C demagnetizes it completely TheCurie temperatures of various ferromagnetic elements and materials areshown in Table 1.1

Each crystal of a ferromagnetic material contains many domains The shape,size, and magnetic orientation of these domains depend on the level anddirection of the applied external field

Let us start with an unmagnetized sample of a ferromagnetic material(Fig 1.7, a) Suppose an external magnetic field Hext in a direction parallel ofthe domain magnetic moments With increasing intensity of the applied field

the domain walls begin to move (wall displacement), first slowly, then quickly,

and at the end, in jumps In the presence of an external field the atomicmagnets are subjected to a torque, which tends to align them with thedirection of the applied field The magnetic moments that are in the direction

of Hext do not experience a resulting torque The magnetic moments that are

not aligned with Hext are subjected to a torque tending to rotate them in the

direction of Hext As a result, the overall domain wall structure becomesmobile and the domains that are in the direction of the applied external field

Hext increase in size by the movement of the domain walls into the domains

with direction opposite to Hext (Fig 1.7b) There will be a net magnetic flux

in the sample The magnetization, which is the average value per unit volume

of all atomic magnets, is increased

When the applied external field Hext is small, the described domain wall

displacements are reversible When Hext is strong, nonelastic wall

displace-ments occur, which cause hysteresis in the B-H relation Above a certain level

of the applied external field, Barkhausen jumps of the domain walls occur

(Fig 1.7c) By these jumps, a domain having the direction of the appliedfield absorbs an adjacent domain with a direction opposite to the appliedfield

When the strength of the applied external field Hext is increased further, the

process of domain rotation occurs The domain magnetic moments rotate in order to align themselves to the direction of Hext, thus increasing the magne-tization The process tends to align the domains more to the direction of theapplied external field in spite of their initial direction along the crystal axes.The total magnetization process includes domain wall displacements andjumps and domain rotations In the case of ferromagnetic metals, at the startthe process is realized mainly by means of the wall displacements and jumps,and the rotations of the whole domains take place at the end of the process,doing the final alignment in the preferred directions, defined by the externalfield

For further reading, the magnetization processes are described in detail instandard texts [1,2]

Let us suppose a magnetic core with a coil, as shown in Fig 1.8 At the

beginning, the net magnetic flux B in the core, the current i in the coil, and the magnetic field intensity H are zero Increasing the current in the coil results in applying the field with intensity H according the Ampere’s law

FIGURE 1.7

Magnetization of a ferromagnetic sample: (a) without applied external field; (b) with applied

external field Hext–movement of the domain walls; (c) with applied external field Hext–rotation

of the domain magnetic moments.

FIGURE 1.8

Magnetic core with a coil.

HextDirection of the

(Hl c= Ni, assuming that H is uniform in the core) The first, slowly rising

initial section of the magnetization curve, Fig 1.9, corresponds to reversibledomain walls displacements In the second section of the curve, the induction

B increases much more quickly with the increase of H and the curve is steep.

The significant increase of B in the second section is explained with the

Barkhausen jumps of the domain walls, which occur when the appliedexternal field intensity reaches a necessary level At the end of this sectionthe structure of the ferromagnetic material contains mainly domains, whichare almost aligned along the crystal axes nearest to the direction of theapplied external field The increase of the magnetic flux in the material is

not any more possible by domain wall motion Further increase in H to larger values results in non-significant increase in B and the third section of the magnetization curve is flat Because the level of H is already much greater

than in section 1 and 2, it is enough to initiate the domain rotation process.The contribution of this process to the total magnetic flux is relatively smalland gradually decreases The material reaches saturation and further

increase in H results in very small increase in B The maximum value of B: the saturation induction value Bsat, is practically reached All the atomic mag-

nets are aligned along the direction of the applied external field H.

Let us observe the process of decreasing H, which means decreasing the excitation current i in the coil The first reaction of reducing H is the rotation

of the domains back to their preferred initial directions in parallel with thecrystal axes Further, some domain walls move back in their initial positions,but most of the domain walls remain in the positions reached in the wall

displacement process Thus, the flux B does not return along the same curve, along which it rises with increasing H The new curve, observed with reduc- ing H, lags behind the initial magnetization curve When H reaches zero,

− Bsat

Bsat

residual flux density or remanence, B r, remains mainly due to non-elastic

wall displacement process To reduce this residual flux density B r to zero, a

negative (reversed) field H is necessary to be applied That field should be

sufficient to restore the initial positions of the domain walls The negative

value of H at which B is reduced to zero is called coercive force or coercivity

of the material H c A further increase of H in the opposite direction results

in a process of magnetization as the one described above and B reaches

saturation level −Bsat, (|−Bsat|= Bsat) If the current of the excitation coil isrepeatedly cycling between the two opposite extreme values, corresponding

to the two opposite maximum values of H, the hysteresis loop is traced out,

The surface of the loop in the B–H plane is the energy loss per volume for

one cycle

According to their coercive force H c the ferromagnetic materials are divided in two general classes:

sub-• Soft magnetic materials

• Hard magnetic materials

Soft magnetic materials are characterized by an ease of change of magnetic

alignment in their structure This fact results in low coercive force H c and anarrow hysteresis loop as shown in Fig 1.11 Soft magnetic materials are ofmain importance for modern electrical engineering and electronics and areindispensable for many devices and applications In power electronics most

of the magnetic components use cores made from soft magnetic materials

Hard magnetic materials are also called permanent magnets The initial

alignment of the magnetic moments in hard magnetic materials strongly resists any influence of an external magnetic field and the coercive force H c

is much higher than that of soft magnetic materials Another importantproperty of permanent magnets is their high value of the remanence induc-

tion B r A typical hysteresis loop of a permanent magnet is shown in Fig 1.11.The permanent magnets produce flux even without any external field Thetypical applications of permanent magnets are in electrical motors, genera-tors, sensing devices, and mechanical holding

The following ranges can be used as approximate criteria for classifying

a material as a soft or hard magnetic material [2]:

H c < 1000 A/m soft magnetic material

H c > 10 000 A/m hard magnetic material

Usually, the values of H c of most of the used in practice materials are

H c< 400 A/m for soft materials and H c> 100,000 A/m for hard magneticmaterials

Permeability is an important property of magnetic materials and therefore wewill discuss it in detail The relative permeability µr introduced in Section 1.1has several different interpretations depending on the specific conditions of

defining and measuring it The index r is omitted and only the corresponding index is used in denoting the different versions: amplitude permeability m a,

initial permeability m i , effective permeability m e , incremental permeability min,

reversible permeability mrev, and complex permeability m.

FIGURE 1.11

Typical hysteresis loops: (a) a sof

magnet-ic material, narrow loop, low H c; (b) a

hard magnetic material, square loop, high

H c and B r.

Hard magnetic

Soft magnetic

B

H

Amplitude permeability m a is the relative permeability under alternating

external field H, which gives the relation between the peak value of the induction B and the magnetic field H Its general definition is

(1.11)

where

is the amplitude induction value averaged out over the core cross-section

is the amplitude field parallel to the surface of the core

The initial permeability m iis the relative permeability of the magnetic

mate-rial when the applied magnetic field H is very low:

(1.12)

For practical purposes the value obtained at a small field H is standardized [2], e.g., as the permeability at H = 0.4 A/m (see Fig 1.12)

If there is an air gap in a closed magnetic circuit, the apparent total

per-meability of the circuit is called effective perper-meability m e, which is much lowerthan the permeability of the same core without an air gap The effective

permeability depends on the initial permeability m iof the magnetic materialand the dimensions of the core and the air gap For cores with relativelysmall (short) air gaps the effective permeability is given by

(1.13)

where

A g is the cross-sectional area of the air gap

l c is the effective length of the magnetic path

FIGURE 1.12

Definition of µi, µ4, and µ ∆ dependent on the

field H.

m m

a o

B H

i A l

g i c

=+

If the air gap is long, some part of the flux passes outside the air gap andthis additional flux results in an increased value of the effective permeability

in comparison with Equation (1.13) Therefore Equation (1.13) is valid onlywhen fringing permeability is neglected The effective permeability is alsoknown as the permeability of an equivalent homogeneous toroidal core

Incremental permeability m∆ is defined when an alternating magnetic field

HAC is superimposed on a static magnetic field HDC The hysteresis loopfollows a minor loop path The incremental permeability is

(1.14)

The limiting value of the incremental permeability min, when the amplitude

of the alternating field excitation HAC is very small, is termed reversible

per-meability mrev (see Fig 1.13):

(1.15)

1.2.4.1 Complex Permeability

In practice, we never have an ideal inductance when the core is made from

a magnetic material Under sinusoidal excitation there is a phase shift

between the fundamental components of the induction B and the magnetic field H By using a complex quantity for the relative permeability, consisting

of a real part and an imaginary part, these effects are easily presented The

imaginary part of the complex permeability µ is associated with the losses inthe material There are two different forms of the complex permeability µ

• Series representation, according to the series equivalent circuit ofmagnetic component shown in Fig 1.14a:

are the real and imaginary parts of the complex permeability

• Parallel representation, according to the parallel equivalent circuitshown in Fig 1.14b):

(1.17)

where

are the real and imaginary parts of the complex permeability

In Fig 1.15 the complex permeability is represented by the series terms inthe frequency domain These values are often given in the data to describe thebehavior of the material at very low induction levels (signal applications) Thegraphs of the real and imaginary parts versus frequency are often shown todescribe the frequency behavior of the material The values of the real andimaginary parts of the complex permeability in the series presentation for a

given frequency can be calculated form the measured inductance L s and

resis-tance R s of the coil of it series equivalent circuit

The parallel representation has the advantage that the loss associated part

does not change when an air gap is added in the magnetic circuit Usually

in applications the induction B is known, which allows the calculation of the

losses directly by using The parallel representation is more often used

in power applications

FIGURE 1.14

Series and parallel equivalent circuits.

FIGURE 1.15

Complex permeability presented by the series

terms in the frequency domain.

Lp

Rp(a)

µ f

f = 0